FROM THE ORDNANCE TRIGONOMETRICAL SURVEY. 
615 
Latitudes and Longitudes. 
For short distances the ordinary formulae are sufficient, but in the case of distances 
above 80 or 100 miles the following formulae are used, A being the given point, and 
B that whose latitude and longitude are required : — 
Let s = distance AB measured on the surface of the earth 
V =normal to minor axis at A 
0 =angle subtended at the foot of this normal by the curve s 
a =azimuth of B at A 
a'=azimuth of A at B, both measured from the north 
K =latitude of A ; »=90—X 
X' = latitude of B 
at =difference of longitude 
§ = radius of curvature of the meridian for the latitude i(>».-|-X') 
*■ 1 / f I y \ sin 0) 1 
tan 2 (a =sin i(jc + fl) 2 “ 
tan 2 (a + ^ + -cosi(ic + 9) 2 ^ 
^ sin !(«' + ? + V 
s . 
’ V ' 6(1— e^) 
cos^ \ cos^ a 
6^92 
^ being a minute angular correction here expressed, as also 0, in angular measure. 
In the calculation of latitudes and longitudes we must suppose all the points to be 
projected on a regular spheroidal surface (A), very nearly agreeing with the actual 
surface covered by the triangulation, then by equations of condition between the 
observed and calculated latitudes, longitudes and azimuths, small alterations to the 
elements of the assumed spheroid and its position must be determined : this new 
spheroid (B) will be that most nearly representing the actual surface under con- 
sideration. 
For the first spheroid, that determined by the Astronomer Royal as most nearly 
representing the earth’s surface ; namely, 
a= 209237 13 feet 
6=20853810 feet 
was usejd as the spheroid of reference (A). 
If we resolve the inclination of the actual surface at any point to that of the sphe- 
roid (B) at the same point, or rather its projection, into two inclinations, one north and 
the other east, and call these inclinations and these quantities being positive 
when the actual surface, as compared with that of the regular surface, rises to the 
MDCCCLVI. 4 M 
